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This problem involves a system with two blocks \( m_1 \) and \( m_2 \), and a pulley of mass \( m_3 \). To solve for the acceleration of the blocks and the pulley, we need to apply Newton's Second Law and account for the fact that the pulley has mass, which means we need to consider rotational dynamics.

### Assumptions:
- The rope is inextensible.
- There is no slipping between the rope and the pulley.
- Friction is ignored on the horizontal surface.

### Forces Acting:
1. **Block \( m_1 \)**:
   - There is a tension force \( T_1 \) acting to the right.
   - No vertical forces since it's on a horizontal surface.
   
2. **Block \( m_2 \)**:
   - The weight of block \( m_2 \) is \( m_2 g \).
   - There is a tension force \( T_2 \) acting upward.

3. **Pulley \( m_3 \)**:
   - The pulley has mass, so its rotational inertia contributes. The torque is generated by the difference in tensions \( T_1 \) and \( T_2 \).

### Equations of Motion:
1. **Block \( m_1 \)**: 
   - \( m_1 a_1 = T_1 \)
   
2. **Block \( m_2 \)**:
   - \( m_2 a_2 = m_2 g - T_2 \)
   
3. **Pulley (rotational dynamics)**:
   - The net torque on the pulley is due to the tension difference:
   - \( (T_1 - T_2) R = I_{\text{pulley}} \alpha \)
   - Where \( I_{\text{pulley}} = \frac{1}{2} m_3 R^2 \) (assuming a solid disk for the pulley).
   - The angular acceleration \( \alpha \) is related to the linear acceleration by \( \alpha = \frac{a_{\text{pulley}}}{R} \).

### Relationships Between Accelerations:
- Because the rope is inextensible, the accelerations are related. If the pulley moves, \( a_1 \) and \( a_2 \) will have a relationship tied to the pulley’s motion. Let's assume:
   - \( a_2 = -a_1 \) (block \( m_2 \) moves down while \( m_1 \) moves to the right).
   - The pulley's linear acceleration \( a_{\text{pulley}} \) might differ from the block's accelerations, but if the pulley moves with an acceleration \( a_{\text{pulley}} \), the accelerations of the blocks are typically related to it.

### Solving the System:
Using the above equations and relations, we can solve for the tensions and accelerations.

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Two massive blocks are shown in the figure below, connected to each other by a rope over a pulley with nonzero mass. Find the acceleration of both masses and of the pulley.

This physics problem involves two massive blocks and a pulley of nonzero mass. Using Newton's Second Law and rotational dynamics, we calculate the acceleration of both blocks and the pulley. The problem includes equations of motion for both blocks and the rotational dynamics of the pulley. Suitable for undergraduate physics students.

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